Positional information and tissue scaling during development and regeneration

In his celebrated 1969 paper, Lewis Wolpert stated that ‘The effective distinguishing feature between mosaic and regulative development is that when a portion of the system is removed, then the mosaic system will largely lack those regions which the removed portion would normally form, whereas in regulative systems a normal pattern would still be formed. I have formalized the problem of the regulative development of axial patterns, whose pattern is size invariant, in terms of the French Flag problem’ (Wolpert, 1969). This statement forms the essence of Wolpert's paper: what is the minimal model that can explain how a pattern forms during development, how does it regulate tissue proportions in differently sized embryos, and how does it lead to regeneration if large portions of tissue are removed from an adult organism? The minimal model that emerged is the famous gradient-based model (often referred to as the ‘French Flag model’) that Wolpert postulated in an earlier paper (Wolpert, 1968): a signaling gradient (e.g. of a diffusible molecule) along a one-dimensional field of elements (e.g. cells) is read out at fixed concentration thresholds that give rise to three different tissue types – in case of the French flag, blue, white and red ones (Fig. 1A).

The idea that gradients underlie patterning had already been expressed in the early 20th century (Boveri, 1901; Morgan, 1901). Subsequently, thresholds were postulated (Dalq, 1938) and Turing coined the term ‘morphogens’ to define molecules responsible for pattern formation (Turing, 1952). Experimental evidence to support the concept that cell fates are induced by gradients of morphogens from signaling centers was provided in the 1960s from transplantation and ablation experiments in moths and leafhoppers (Sander, 1959; Stumpf, 1966). This led to the idea that each cell can be assigned a positional value relative to a signaling center (Lawrence, 1966; Rogers and Schier, 2011; Stumpf, 1966; Wolpert, 1968, 1969).

If in such a scenario the gradients or thresholds do not change concomitantly with tissue size, incorrect tissue proportions should arise (Fig. 1B,C). However, this is not observed in the vast majority of embryos analyzed so far. Instead there are numerous examples from development and regeneration throughout the animal kingdom of proper patterning in differently sized individuals. This raises the intriguing issue of how signaling gradients scale in order to give rise to the correct tissue dimensions. Patterning mechanisms must not only convey positional information, but they also have to be robust against changes in the size of the patterning field.

In this Review, we discuss mechanisms that provide size invariance of patterning systems. We focus on scaling systems during animal development, when positional information is established de novo, and in adult regeneration, when positional information must be re-established and reconnected to the polarity of existing structures. We highlight mathematical models that describe scaling, illustrate mechanisms that allow the scaling of morphogen gradients and provide examples of experimental data supporting these concepts.

To provide historical context for our analyses, we first provide an overview of classical experiments with sea urchins, Hydra, fish and amphibians that have demonstrated that correct tissue proportions can be established even when the overall size of an organism is dramatically reduced after surgical manipulation (Cooke, 1975; Driesch, 1892; Hoadley, 1928; Holtfreter, 1938; Mangold, 1960; Morgan, 1895; Nicholas and Oppenheimer, 1942; Sander, 1959; Spemann, 1938; Wolpert, 1969).

The issue of how robust development is towards size manipulation arose in the late 19th century, originally as a tangent to the debate on preformationism. In his Princip der organbildenden Keimbezirke, Wilhelm His proposed that the germ disc of chicken embryos contains the primordia for all organs as flat projections and that, vice versa, each position on the germ disc corresponds to a specific position in a later organ (His, 1874). To investigate this hypothesis, Wilhelm Roux performed experiments in which he killed one blastomere of amphibian embryos at the two-cell stage with a hot needle (Box 1) (Roux, 1885). He found that this treatment led to the development of half embryos – embryos in which the body parts that would have been formed by the killed blastomere are missing – that he dubbed ‘Halbbildungen’. This was interpreted as a confirmation of the preformationism theory, as half embryos could not establish their full body plan.

Roux's results were later challenged by Hans Driesch, who separated blastomeres of sea urchin embryos through shaking and found that these embryos developed into intact smaller versions (‘Theilbildungen’) of their untreated siblings, contradicting Roux's results (Driesch, 1892). Based on these and the earlier findings of Chabry (1887) and the Hertwig brothers (Hertwig, 1890; Hertwig and Hertwig, 1887), Driesch concluded that embryos are not predetermined. However, he did not comment on potential mechanisms underlying the observed scale-invariant patterning.

When Thomas Hunt Morgan re-examined the results of Hertwig and Roux, he found that both datasets also contained cases of the other, i.e. Hertwig also obtained a small proportion of Halbbildungen and Roux's method generated some size-reduced whole embryos (Hertwig, 1893; Morgan, 1895; Roux, 1894). When Morgan repeated the experiments on frog embryos, he likewise found both types of experimental outcomes. Eventually, the outcome of these experiments turned out to depend on an additional factor, namely whether the killed cell was removed or not (Bruns, 1931; Vogt and Bruns, 1930). Morgan eventually determined that scaling must take place in those instances in which viable adults developed: ‘Readjustments to the size of the mass must take place because the embryo that emerges is not the anterior half of an embryo but a whole embryo of half size’ (Morgan, 1924).

An increasing number of examples of embryos that can form perfectly scaled, albeit smaller, individuals after the separation of blastomeres was subsequently found across different species, e.g. lancelets and newt (Morgan, 1896; Wilson, 1893). Moreover, experiments by Mangold and Spemann as well as Holtfreter showed more specifically that individual structures can scale with embryo size and that substructures, such as the lateral part (Seitenkappe) of an amphibian gastrula, can give rise to a full bilateral embryo (Holtfreter, 1938; Ruud and Spemann, 1922; Spemann and Mangold, 1924). However, the discussion still revolved around the problem of preformation rather than how scale invariance is achieved.

The question regarding the molecular mechanisms of scale-invariant patterning only moved into the spotlight with the introduction of mathematical models of development, most notably Wolpert's gradient model described above, and Turing's and Gierer and Meinhardt's reaction-diffusion (RD) systems (Gierer and Meinhardt, 1972; Turing, 1952; Wolpert, 1968, 1969). These RD systems are based on two diffusible components. One molecule, termed the ‘activator’, positively feeds back on its own synthesis and at the same time activates the other component (Gierer and Meinhardt, 1972). The other molecule is termed the ‘inhibitor’ because it negatively regulates itself and the activator. Indeed, it has since been shown that interactions between diffusible activators and inhibitors can amplify small random fluctuations, leading to self-organized repeated wave-like patterns of signaling that can be translated into periodic morphological elements (Landge et al., 2019).

Although both the gradient and RD models can explain many instances of patterning, it became clear that they are not sufficient to fully account for scale-invariant development. In 1975, Cooke counted the somites of surgically size-reduced Xenopus embryos and found that somite size, but not somite number, was changed (Cooke, 1975). He pointed out that a Turing-type RD system should give the opposite outcome, unless the physiochemical properties of the involved molecules, such as their diffusivities, are changed. Pure positional information models, on the other hand, would not easily give rise to repeated elements. He subsequently developed the clock-wavefront model of somitogenesis, which incorporated a scale-invariant gradient of positional information and a cellular oscillator for generating repeated elements (Cooke and Zeeman, 1976).

It also became clear that there are limits to the scaling capacity of embryos. As mentioned above, embryos do not always downscale and compensate if the dead cell is not removed (Bruns, 1931; Vogt and Bruns, 1930). Likewise, the time window during which an organism can adjust to achieve normal patterning can be limited. For example, Klaus Sander's Schnürungs experiments, in which he divided embryos of the leafhopper Euscelis using a nylon loop (Box 1), revealed that scaling is not only dependent on the position of missing body parts but also depends on the developmental stage during which the procedure is performed (Sander, 1959). Similarly, manipulated amphibian gastrulae can form complete body plans, whereas surgery at later neurula stages results in missing body parts (Mangold, 1960). Finally, although it remains unclear how widespread scaling actually is, it is likely that most animals with a regulative mode of embryogenesis possess the ability to scale tissue proportions to some extent (while organisms that exhibit mosaic development, such as nematodes, cannot compensate for the loss of blastomeres).

In order for scaling to occur, information about tissue size must be encoded in some aspect of the patterning system. Below, we outline the mathematics and underlying concepts of four major experimentally supported scale-invariant patterning models.

This equation demonstrates perfect scaling of the gradient that was generated by this mechanism, as the spatial profile of the morphogen M is a function of relative position x/L rather than x alone (Ben-Zvi and Barkai, 2010). Crick found that relevant gradients in tissues using this mechanism could form over biologically realistic time scales, demonstrating the plausibility of a gradient formation mechanism (Crick, 1970). However, how the morphogen concentration at the ends of the field could be held constant, as postulated by Wolpert, remained unresolved.

If the size of the source is changed in differently sized embryos, this would automatically change the production term j_M. For example, a smaller source would provide a smaller flux j_M, whereas a larger source would provide a higher flux j_M. This change in the flux would change the amplitude and the range (Fig. 2A, bottom panel), but it does not yield perfect scaling of the morphogen gradient at all positions of the patterning field (Umulis and Othmer, 2013). However, this approximate scaling is likely sufficient in many contexts, especially if the gradient provides a rough pre-pattern that is later refined by other patterning systems.

Therefore, a simple mechanism for adjusting a gradient to a new size is to invoke a change in the boundary conditions. In this model, if the target tissue width becomes reduced, the production rate must decrease accordingly in order to maintain the relative shape of the gradient. If, on the other hand, the tissue grows, the production must increase to match the wider patterning field (Fig. 2A).

This relationship between λ, D_M and k_M implies that the range of a gradient could be adjusted by changing the diffusion coefficient D_M or the degradation constant k_M of the morphogen. Therefore, a good strategy to achieve scale invariance is to couple these two biophysical properties to field size. From first principles, however, it is difficult to conceive how the biophysical properties of a morphogen could be influenced by a change in the size of the patterning field. According to the Einstein-Stokes relation, the diffusion coefficient D_M depends on the size of the morphogen, the viscosity of the medium in which it diffuses and the surrounding temperature (reviewed in Müller and Schier, 2011). All of these properties should in principal be unaffected in differently sized patterning fields, and therefore the diffusion coefficient would remain unchanged.

This equation shows perfect scaling of the morphogen gradient, as the spatial profile of M is a function of relative position x/L rather than x alone (i.e. as seen in the simple system without a modulator described above). In this model, the modulator needs to have a high diffusivity D_s, so that it can relay information about field size based on its accumulation in smaller domains or dilution in larger domains. In smaller embryos, the concentration of the highly diffusible inhibitor would rise (Fig. 2B) and therefore shrink the range of the morphogen. In larger embryos, the inhibitor would be more diluted because of its high diffusivity, allowing the morphogen to disperse over a wider region (Fig. 2B).

Modulator molecules whose expression is independent of the morphogen have been called passive modulators, whereas modulators whose expression is strongly connected to morphogen concentration have been called active modulators (Umulis and Othmer, 2013). Conceptually similar models with different boundary conditions that lead to self-organized and size-independent pattern formation include wave pinning (Ishihara and Tanaka, 2018; Mori et al., 2008) and models with saturation of autocatalysis (Gierer and Meinhardt, 1972). Related models in which morphogen diffusivities are proportional to field size have also been proposed (Othmer and Pate, 1980; Pate and Othmer, 1984).

This mechanism based on two opposing morphogen gradients works well to scale the boundaries of individual target genes, but it only poorly accounts for the scaling of multiple morphogen-dependent responses with different activation thresholds, as the readout gradient does not scale well across the entire field (McHale et al., 2006). The degree of scaling for opposing gradient-based mechanisms can be improved when the molecules are irreversibly inactivated upon binding to each other (McHale et al., 2006).

Interestingly, the mirror-symmetric ‘induction-contraction’ model – in which a contractor molecule is activated by the morphogen and contracts its range – also scales (Fig. 2D, bottom panel) (Rahimi et al., 2016), and similar principles can even be applied to self-organizing pattern formation systems that lack initial polarity (Werner et al., 2015). Finally, feedback between tissue mechanics and morphogen turnover or transport can also lead to size-invariant scaling in advection-driven self-organizing models, even with a single morphogen species (Recho et al., 2019).

In summary, tissues that depend on precise morphogen gradients need to adjust the length scale of morphogen distribution to the size of the tissue, which can be achieved by special boundary conditions, opposing gradients and modulators affecting the mobility or stability of the morphogen. It is possible that different strategies or combinations of strategies are used in different developmental contexts, and it is therefore necessary to perform experimental measurements for model validation. Finally, it will be important to extend future theoretical models to take into account the noise and variability that is often observed in patterning systems.

Owing to advances in molecular biology, imaging, genomics and computational biology, the last 15 years have seen a surge of scaling studies that have uncovered some of the mechanisms underlying the observations of classical experimental embryology described above. Below, we highlight recent studies that have used modern molecular biology methods to identify new principles controlling scale-invariant patterning during embryogenesis and regeneration. We analyze these findings in the context of the concepts and mathematical models introduced above.

Numerous experiments have demonstrated the existence of mechanisms that allow embryonic axes and tissues to scale during development. For example, surgically reducing the size of developing zebrafish embryos by 30% (Box 1) typically results in normally patterned, viable larvae and adults (Almuedo-Castillo et al., 2018; Collins et al., 2019 preprint; Huang and Umulis, 2019; Ishimatsu et al., 2019, 2018). Even more drastically, amphibian half embryos (Box 1) that originate from either the lateral (Moriyama and De Robertis, 2018; Reversade and De Robertis, 2005) or dorsal half (Spemann, 1903) form properly patterned miniature organisms. Here, we focus on some well-studied vertebrate systems and synthesize the mechanisms that can explain classical embryological experiments, from Spemann's early dorsal-ventral patterning experiments to Cooke's somite scaling observations. Excellent progress has also been made in invertebrate systems, as reviewed previously (Umulis and Othmer, 2013).

The dorsal-ventral (DV) axis of vertebrate embryos is patterned by a gradient of bone morphogenetic proteins (BMPs) that activate the phosphorylation of the transcriptional effector Smad1/5/9 (pSmad1/5/9) via heterodimeric kinase receptors, leading to pSmad1/5/9-mediated induction of BMP target genes. During the blastula and early gastrula stages of vertebrate development, BMP expression peaks ventrally, while expression of the BMP antagonist Chordin is maximal dorsally. Loss of BMP signaling produces dorsalized embryos, while gain of function leads to ventralized phenotypes in a dose-dependent manner, highlighting that tight regulation of the BMP pathway is necessary (reviewed by Rogers and Müller, 2019).

Following bisection, the lateral halves of a Xenopus blastula close the wound such that the original dorsal and ventral signaling centers become juxtaposed, allowing the halved embryos to give rise to two properly patterned smaller tadpoles (Fig. 3A) (Moriyama and De Robertis, 2018). Both centers shift by 90°, eventually ending up on opposing sides of the embryo, as shown by pSmad1/5/9 and Chordin staining (Moriyama and De Robertis, 2018). Another BMP ligand, ADMP (anti-dorsalizing morphogenetic protein), is expressed at low levels and represses its own expression via positive feedback on BMP, thereby establishing a self-regulatory system (Reversade and De Robertis, 2005). ADMP is indispensable in this system, as the combination of BMP and ADMP determines the shape of the BMP activity gradient (Ben-Zvi et al., 2011b; Reversade and De Robertis, 2005). Facilitated diffusion of BMP and ADMP upon binding to Chordin has been suggested to effectively ‘shuttle’ BMP activity towards the ventral side away from the Chordin source, which would allow this system to adapt to different embryo sizes (Fig. 3B) (Ben-Zvi et al., 2008). It has therefore been suggested that this network comprises an expansion-repression system (discussed above) for scale-invariant patterning, in which ADMP functions as the expander that is repressed by BMP.

As an alternative, a long-range accumulation and feedback model has recently been suggested (Fig. 3C) (Inomata et al., 2013). This model is based on data showing that the highly diffusible BMP target Sizzled stabilizes Chordin levels by inhibiting its protease-mediated degradation. This in turn feeds back on BMP activity and thus on sizzled expression, thereby forming an embryo-wide feedback loop that provides size invariance (Inomata, 2017; Inomata et al., 2013). It is likely that both Sizzled and ADMP act as expanders, and it is possible that they together control robust scaling of DV patterning during Xenopus development through a ‘double-expander’ mechanism (Fig. 3D) (Ben-Zvi et al., 2014).

Immunostaining for pSmad1/5/9 has revealed that a scaled gradient of BMP signaling also accounts for the size invariance of surgically modified zebrafish embryos (Fig. 3E,F) (Huang and Umulis, 2019). A shuttling mechanism in zebrafish, as proposed for Xenopus, would require that: (1) the diffusivity of BMP is much smaller than that of Chordin; (2) Chordin enhances BMP diffusion; (3) the half-life of BMP is larger than that of Chordin; and (4) the ventral pSmad1/5/9 peak decreases in the absence of Chordin. However, all of these conditions have been refuted experimentally (Pomreinke et al., 2017; Zinski et al., 2017). The mechanism underlying zebrafish scale-invariant DV patterning is therefore currently unclear, but it is possible that it is regulated by the concentration of one of the BMP signaling inhibitors that might act as a scaling modulator.

During gastrulation, the inner-to-outer body organization is determined and embryos specify the correct proportions of cells that contribute to the three germ layers: endoderm, mesoderm and ectoderm (Fig. 4A). These decisions are regulated by an activator-inhibitor system comprising Nodal signaling molecules, which induce mesendodermal cell fates, and Lefty proteins, which antagonize Nodal proteins and thus promote ectodermal fates (reviewed by Rogers and Müller, 2019; Schier, 2009). Computational screening and in vivo experiments have revealed that the concentration of the highly diffusive Nodal inhibitor Lefty acts as a size sensor (Fig. 4B): after removal of 30% of the cells from the embryonic animal pole (Box 1), the short-range Nodal gradient remains largely unaffected, whereas the highly diffusive inhibitor Lefty is reflected from the new boundary and fills up the smaller patterning field (Almuedo-Castillo et al., 2018). Thus, the domain of Nodal signaling and hence mesendodermal target gene expression shrinks to re-establish the correct germ layer proportions. As differential diffusivity of activators and inhibitors is essential for patterning in this system (Müller et al., 2012), embryos do not scale if Lefty diffusion is inhibited (e.g. by tethering Lefty to membranes; Fig. 4C) (Almuedo-Castillo et al., 2018). Strikingly, lefty mutants can be rescued by exposure to ectopic small-molecule Nodal inhibitors (Rogers et al., 2017). Surgically shortened lefty mutants can also be rescued by inhibitor exposure, but they require higher concentrations than unmanipulated mutants (Fig. 4D) (Almuedo-Castillo et al., 2018), demonstrating that appropriate size-dependent inhibition levels are required to achieve the correct germ layer proportions.

Conceptually, germ layer scaling is achieved through a mechanism involving a size-dependent highly mobile modulator, as introduced earlier. As lefty expression is under the control of Nodal signaling (reviewed by Rogers and Müller, 2019), this system can superficially be interpreted as an active modulator or induction-contraction network (Nesterenko and Zaraisky, 2019). However, Lefty-mediated size control in zebrafish embryos can also be uncoupled from Nodal signaling by creating artificial Lefty sources that are not part of a feedback mechanism (Almuedo-Castillo et al., 2018), showing that a passive modulator system is sufficient to explain scale-invariant patterning in this context.

The vertebrate neural tube is patterned into 13 populations of different neuronal progenitors along the DV axis (Alaynick et al., 2011). This pattern is regulated by two signaling centers: the dorsal roof plate, which secretes BMP and Wnt signals, and the floor plate and notochord, which secrete Sonic hedgehog (Shh) (Fig. 5A, top) (Briscoe et al., 2001; Liem et al., 2000, 1997; Roelink et al., 1995). Thus, the neural tube is patterned by opposing gradients of signaling molecules. However, cells do not seem to measure the ratio between BMP and Shh, as would be the case for an opposing gradient model, as cells that are exposed to high levels of both signals adopt either dorsal or ventral rather than intermediate fates (Collins et al., 2019 preprint; Zagorski et al., 2017).

During development, the progenitor domains in the neural tube undergo changes in size as the tissue grows (Box 2). Unlike other patterning systems, the morphogen gradients do not scale during growth; after the initial cell fate specification phase, cell type-specific differentiation dynamics control domain proportions (Fig. 5A) (Kicheva et al., 2014). This has been elegantly demonstrated in studies of mice that are heterozygous for a deletion in rpl24, which encodes a large ribosomal subunit. These mice have about 20% smaller neural tubes, but they exhibit the same tissue proportions as wild-type mice (Fig. 5A, bottom) (Kicheva et al., 2014). How size feeds back on differentiation rate is not precisely understood, but it has been suggested that post-mitotic neurons are involved (Kicheva et al., 2014).

A different mechanism has been found in artificially size-reduced fish embryos. Here, a variant of the expansion-repression model is thought to scale the ventral Shh gradient via the secreted matrix protein Scube2 (Fig. 5B) (Collins et al., 2019 preprint; Shilo and Barkai, 2017). Scube2 displays the classical features of an expander: it is repressed by Shh signaling, and it cell non-autonomously enhances the range of Shh signaling (Collins et al., 2019 preprint). How the dorsal BMP gradient adjusts its size during neural tube patterning is not yet known, but it is tempting to speculate that a similar expansion-repression mechanism might act in this context, as the secreted Chordin stabilizer Sizzled (and potentially admp in species that possess this gene, such as zebrafish) enables the early embryonic BMP gradient to scale in an expander-like manner (Ben-Zvi and Barkai, 2010; Ben-Zvi et al., 2014, 2008; Inomata et al., 2013).

During development, somites are formed from the presomitic mesoderm (PSM) in an anterior-to-posterior sequence. This process can be described by a clock-wavefront model (Cooke and Zeeman, 1976; reviewed by Oates et al., 2012). In this model, tissue polarity is controlled by posterior-to-anterior signaling gradients of FGFs and Wnts, and an opposing gradient of retinoic acid (Fig. 6A) (Aulehla and Pourquié, 2010). The frequency of somite formation is controlled by a clock, which consists of a system of coordinated oscillators in PSM cells that exhibits species-specific timing. For example, in zebrafish a somite is formed every 25 min, in frogs or chicken every 90 min, and in mouse every 2 h (Gomez et al., 2008; Schröter et al., 2008). The position of somite formation is determined by a travelling wavefront that freezes the oscillating cells and thereby causes formation of a new somite. The main oscillator genes are orthologs of the hairy and enhancer of split family (her or hes) as well as components of the Wnt, FGF and Delta/Notch pathways (Oates et al., 2012; Palmeirim et al., 1997).

As mentioned above, the first evidence to suggest that somite size scales with body size was provided by Cooke in 1975 (Cooke, 1975), but the mechanism underlying this scaling remained unclear for a long time. Conceivably, four factors could contribute to the size-invariance of somite patterning: the clock period, the axis elongation speed, the wavelength or the gradients of the wavefront (Ishimatsu et al., 2019, 2018). PSM cells are known to proliferate, but more cells are lost to the forming somites, and over time the PSM shrinks. However, it is not entirely clear whether PSM length and the length of the formed somites are correlated (Cooke, 1975; Gomez et al., 2008; Ishimatsu et al., 2018; Lauschke et al., 2013). Changes in the period of the segmentation clock have been shown to change somite length, but this does not scale with embryo length during normal development (Harima et al., 2013; Herrgen et al., 2010; Schröter and Oates, 2010). Likewise, studies in surgically size-reduced zebrafish embryos have shown that neither the clock period nor the axis elongation speed scale in this case (Ishimatsu et al., 2018).

What, then, does control scaling in this context? The wavelength of the her1 gene does not scale with normal PSM size reduction during development, but it does in size-decreased embryos (Ishimatsu et al., 2018). In addition, immunohistochemistry for the FGF effector ERK, together with FRET-sensor experiments, have revealed that the FGF gradient scales both during normal development and after surgery (Fig. 6B,C). Thus, a suggested mechanism for this ‘clock and scaled gradient’ system involves negative feedback of newly formed somites on FGF that leads to gradient shortening. The precise molecular mechanism is not yet clear, but multiple options come to mind. The opposing gradients of Wnt/FGF and somite-derived retinoic acid could constitute an opposing gradient system that allows scaling (Oates et al., 2012). Alternatively, feedback-based mechanisms could control somite size, as Wnt and FGFs have been shown to act in auto-regulatory feedback loops in other systems (e.g. during regeneration, discussed below).

Varying degrees of regenerative capacity are found in almost all animals. Amphibians have remarkable regenerative abilities that enable them to regrow an entire limb or reconnect a severed spinal cord. Cnidarians (such as Hydra) and planarians (such as Schmidtea mediterranea) are even capable of whole-body regeneration (reviewed by Ivankovic et al., 2019; Joven et al., 2019; Vogg et al., 2019b). During regeneration, animals recapitulate various embryonic processes and developmental pathways, indicating that scaling of positional information is also necessary. Importantly, regenerating tissues must grow to the size of an adult animal. This process appears even more complex than embryonic size invariance, as signaling must first scale to accommodate the reduced body size and, subsequently, rescaling during growth must be accomplished (Box 2). Additionally, regeneration does not have a fixed starting point but is initiated at a random point of wounding, suggesting the need for a self-organizing system (Stückemann et al., 2017). In this section, we discuss mechanisms of size control in Hydra (Vogg et al., 2019a), planarians (Stückemann et al., 2017; Werner et al., 2015) and amphibians (Bryant et al., 2017; Vincent et al., 2015) from the perspective of positional information.

The freshwater polyp Hydra is capable of remarkably size-invariant regeneration. As Lewis Wolpert stated in his 1969 publication: ‘a fragment of hydra, one-hundredth its volume can give rise to an almost complete animal’ (Wolpert, 1969). The Hydra body consists of an apical head structure with tentacles and a dome surrounding the mouth, called a hypostome, a body column and a basal foot structure (Fig. 7A). Transplantation experiments conducted more than 100 years ago revealed that the hypostome has inducing capacity similar to what was later termed an embryonic organizer (Browne, 1909). Adult Hydra polyps retain a permanent stem cell population, and axial polarity is maintained by signals from the hypostome (Hobmayer et al., 2000). If the head and foot parts of the animal are removed, the gastric section regenerates and maintains the original polarity (Wilby and Webster, 1970). After mid-gastric bisection, both pieces regenerate the missing body parts, with head-organizer activity being established after 3 h in the head-regenerating tip and retained for about 2 days (MacWilliams, 1983).

An RD system has been proposed to explain the inductive abilities of the head organizer and the transplantation and regeneration data (Gierer and Meinhardt, 1972). In this model, the head region contains high levels of an activator that, together with its long-range inhibitor, is thought to form a gradient across the body axis. In a system like this, any part of the gastric region will retain its polarity even if the gradient is shallow (Gierer and Meinhardt, 1972).

A number of molecules that might be involved in this activator-inhibitor system have been identified. The role of the activator, a promoter of head identity, is provided by Wnt signaling (Hobmayer et al., 2000), and the transcription factor HySp5 has recently been identified as the inhibitor of the system (Vogg et al., 2019a). HySp5 is a target gene of Wnt signaling but, in contrast to the predictions of the RD theory, it acts cell-autonomously and is not diffusible (Vogg et al., 2019a). It has been shown, however, that multi-component RD systems can also comprise non-diffusible locally acting elements (Marcon et al., 2016). HySp5 could therefore act as a spatially fixed capacitor that might integrate other diffusible signals (Landge et al., 2019; Marcon et al., 2016).

Owing to their original axial polarity, head and foot regeneration differ to some extent. After mid-body section, the formation of a new organizer in the head-regenerating tip is likely triggered by elevated Wnt levels in direct response to wounding via apoptosis (Chera et al., 2009; Galliot, 2013). After this initial Wnt burst, from day 2 onwards, these signals re-establish the gradient along the body axis (Gufler et al., 2018; MacWilliams, 1983). Models indicate that, by removing the area of high activator (i.e. the head part), the remaining inhibitor decays and a new maximum can arise due to fluctuations in the source cells – in this case the apoptotic Wnt burst – or by retaining the original polarity based on relative activator levels (Fig. 7A) (Gierer and Meinhardt, 1972; Meinhardt, 2009).

Regeneration of the Hydra foot, by contrast, is thought to resemble a wound healing response (Galliot, 2013). Therefore, foot regeneration does not require additional positional information: a new foot structure can be generated wherever the activator-to-inhibitor ratio is lowest. This gradient scales to the new body size, because the remaining inhibitor has a smaller field in which to disperse and thus the activation is reduced (Fig. 7A) (Meinhardt, 2009). Thus, this system displays properties consistent with the modulator models discussed earlier.

Many planarians, such as Schmidtea mediterranea or Planaria maculata, can regenerate full animals after being cut into pieces, and even one 279th of a dissected planarian flatworm can give rise to a normally patterned individual over time (Morgan, 1898). The workhorse of planarian regeneration is a population of embryonic-like stem cells termed neoblasts that are found throughout the adult body (reviewed by Reddien, 2018). Positional information during homeostasis is maintained by muscle cells (Witchley et al., 2013). After transversal mid-body bisection, the first challenge is to re-establish proper anterior-posterior polarity. An activator-inhibitor Turing-like RD system has been suggested to pattern the animal during development and regeneration (Werner et al., 2015), but Turing models in their simplest form do not scale and create repeated patterns (Müller and Nüsslein-Volhard, 2016; Murray, 2013). If this was the patterning mechanism, a flatworm could end up with two heads or tails (Fig. 7B) (Cooke, 1975; Werner et al., 2015). However, if an additional molecule – an expander – is introduced, the system can spontaneously self-organize and also scale (Werner et al., 2015). In this system, the expander must negatively regulate both the activator and the inhibitor, and it must be degraded by the inhibitor. Another explanation for the re-establishment of anterior-posterior polarity in planarian regeneration, supported by the shape of signaling gradients in double-tailed individuals, is that two signaling centers exist at opposing poles, thus forming part of a ‘dual gradients’ model (Fig. 7C,D) (Stückemann et al., 2017).

Like the head organizer of Hydra, the planarian posterior signaling center comprises a Wnt feedback loop; accordingly, Wnt inhibition forces head formation and inhibits tail regeneration (Gurley et al., 2008; Iglesias et al., 2008; Petersen and Reddien, 2008). Consequently, the levels of Wnt signaling increase or decrease at the wound sites of tail or head regenerating pieces, respectively, within the first 24 h and return to their normal levels within 7 days (Fig. 7E,F) (Gurley et al., 2010; Petersen and Reddien, 2009; Stückemann et al., 2017). This is thought to be caused by a generic wound response mediated by a wnt1/notum switch that represses Wnt-dependent tail regeneration in anterior-facing wounds (Fig. 7E,F) (Gurley et al., 2010; Petersen and Reddien, 2009, 2011; Stückemann et al., 2017; Wenemoser et al., 2012; Wurtzel et al., 2015). In agreement with both the expander (Werner et al., 2015) and the dual gradients model (Stückemann et al., 2017), uniformly high levels of β-catenin cause uniform expression of tail genes. Knockdown of the Wnt pathway, however, induces the formation of additional heads along the body axis that are properly patterned, supporting the dual gradient model (Iglesias et al., 2008; Stückemann et al., 2017).

If the dual gradients model holds true, what might be the nature of the signal that opposes the Wnt gradient? Studies have shown that a gradient of ERK signaling that is maximal anteriorly (Fig. 7C) could fulfill the role as the opposing gradient to the posterior-to-anterior Wnt gradient (Agata et al., 2014; Tasaki et al., 2011; Umesono et al., 2013). Likewise, Hh signaling has a regulating effect on Wnt signaling, thereby contributing to polarity (Rink et al., 2009; Yazawa et al., 2009).

Although a system of opposing gradients would be able to scale by itself, computational analyses have suggested that the active transport of morphogens can aid scaling (Pietak et al., 2019). In this model, Hedgehog molecules are transported along axons by kinesin in an anterior-to-posterior direction, and a notum-inducing factor is subject to dynein transport in the other direction. In this way, morphogen gradients form relative to the direction of the neuronal vector field independent of tissue size and symmetry (Pietak et al., 2019). Thus, three different models are currently able to explain polarity re-establishment in regenerating planarians, and more work is required to unify these theoretical frameworks.

Regeneration in salamanders is initiated after wound closure by the accumulation and proliferation of diverse mesenchymal cells that form the blastema: a region of undifferentiated cells that is crucial for regeneration. Proliferation and subsequent differentiation of these cells into tail elements (myotomes) or limb elements then proceeds until the right amount of limb or tail is regrown. After distal limb amputation, a hand is regenerated, while a more-proximal injury leads to the formation of a whole arm. However, blastema cells from the hand are not able to regenerate a whole arm when transplanted into a more proximal position, suggesting that the original positional information of the tissue is retained during regeneration (Pescitelli and Stocum, 1980; Roensch et al., 2013). Interestingly, the duration of the regeneration phase is independent of the size of the structure. In the first phase of tail regeneration (15 days after amputation), for example, blastema cells are in a maximum growth phase, independent of their proximal-distal position. However, proximal cells differentiate at a lower rate while blastema cells continue proliferating. This surplus of stem cells is thought to cause the faster growth observed after proximal injuries compared with distal injuries (Bryant and Gardiner, 2016; Vincent et al., 2015). Therefore, the amount of tissue that is regenerated appears to be controlled by cell-cycle gradients that in turn are thought to be regulated by morphogen gradients (Bryant and Gardiner, 2016).

Strikingly, regeneration does not occur if tissues of the same polarity are in contact. If, for example, an anterior part is grafted to replace the posterior part and thus a double anterior limb is created, regeneration cannot proceed. On the contrary, supernumerary limbs are produced if the polarity is surgically inverted, such that multiple juxtapositions of anterior and posterior tissues occur (Fig. 7G) (Butler, 1955; Iten and Bryant, 1975). This suggests that whenever anterior blastema cells are positioned next to posterior ones, a signaling center that promotes growth is formed (Tanaka, 2016).

The molecular network underlying this phenomenon consists of anterior FGF8, posterior Shh and the BMP inhibitor Gremlin (Grem) (Fig. 7H) (Nacu et al., 2016). Additional proximal-distal positional information is provided by a retinoic acid gradient (McCusker et al., 2014), with connective tissue being the primary responsive cell type (Kragl et al., 2009; Nacu et al., 2013). The FGF-Shh system is reminiscent of that observed during embryonic limb development, where growth is maintained by FGF8 signaling from apical tissue, which in turn requires Shh signaling from the posterior zone of polarizing activity (ZPA) in order to be maintained (Chiang et al., 2001) (for a review of vertebrate limb development, see Saxena et al., 2016). In this system, Grem functions as a relay between Shh and FGF signaling, and growth terminates once a population of cells unable to express grem is established (Scherz et al., 2004). But how, if at all, does this system scale? Whereas during embryogenesis the tissue is patterned while it grows, the field size during regeneration is that of an adult limb. It is therefore likely that the FGF, Grem and Shh gradients do scale in this context, although this has not yet been experimentally demonstrated (Tanaka, 2018). A version of the expander mechanism for scaling would be attractive: in this scenario, mesenchymal BMP signaling inhibits FGF signaling from the anterior mesenchyme, and BMP in turn is inhibited by Grem, which depends on Shh signaling from the ZPA. This relay could be used to measure the distance between the anterior mesenchyme and the ZPA and thus field size. It should also be noted that the limb patterning network in axolotls is based on FGF, Shh and RA, but also on canonical Wnt signaling, as regeneration does not occur if Wnt signaling is inhibited (Kawakami et al., 2006). Therefore, it is tempting to speculate that this might be homologous to the first position-independent phase of wnt expression in Hydra regeneration (Gufler et al., 2018) and the generic wound response in planarians (Wenemoser et al., 2012; Wurtzel et al., 2015).

Overall, these studies indicate that a number of molecular pathways could together provide scaling, but more data are required before speculations about their mechanisms of action can be converted into useful hypotheses. For example, visualization of the signaling molecules involved is needed to clarify whether these form gradients and whether those gradients scale. Additionally, knockdown and uniform rescue experiments could provide information as to whether gradients are required for regeneration.

Over the past 50 years, Lewis Wolpert's concept of positional information (Wolpert, 1968, 1969) has been fundamental for the field of developmental biology. A major motivation for Wolpert's positional information concept was the need for a mechanism to explain how differently sized embryos scale their tissue proportions. The simple gradients that Wolpert proposed have subsequently inspired new theories, and recent experimental and theoretical advances over the last 15 years have allowed direct tests of these ideas.

Four major models have been proposed that can account for scale-invariant patterning: (1) scaling by specific boundary conditions; (2) modulating signaling gradients by highly diffusible molecules; (3) using opposing gradients; and (4) feedback-based systems. There is no reason to believe that only one of these models is applicable to every case. On the contrary, it appears that even the same signaling pathways can scale using different strategies. Shh, for example, is size adjusted in the zebrafish neural tube via a feedback mechanism, while its size invariance in avian interspecies scaling (Box 3) is based on differential target tissue responsiveness. With more experimental data from different systems, our understanding of the general mechanisms of scaling will expand.

Currently many scaling scenarios are only partially understood. The BMP gradient in zebrafish DV patterning, for example, has been shown to scale, but the underlying mechanism is unclear. A shuttling function, as suggested for Xenopus, has been ruled out. It is tempting to speculate that BMP signaling scales analogously to the Nodal pathway, with Sizzled playing the role of Lefty, because Sizzled is diffusible, negatively feeds back on BMP, and is a strong target of BMP signaling, just like Lefty for Nodal (Dubrulle et al., 2015). Interestingly, Xenopus BMP signaling and zebrafish Nodal signaling appear to use rather different mechanisms for scale-invariant patterning, although both ligands belong to the TGFβ superfamily and thus likely have similar properties. Therefore, analyses of amphibian germ layer and fish DV patterning hold the potential to greatly advance our understanding of scaling and patterning mechanisms in general.

The similarities and parallels between development and regeneration are striking. Molecules that are used for the initial patterning during development are also re-deployed during regeneration. The Wnt pathway, for example, is crucial for determining anterior-posterior polarity in both development and regeneration, and in the case of planarians and cnidarians also in maintaining this polarity once a steady state is reached. Furthermore, there are strong resemblances mechanistically, such that the dilution of inhibitors or expanders in both contexts can act to scale tissue proportions.

Moving forward, it will be interesting to see many of the gaps in our knowledge close and new approaches emerge. For example, most studies have focused on artificially reducing the size of embryos, but it would also be illuminating to increase embryo size in order to determine the limits of scaling. Intra-species experiments also have the potential to not only broaden our understanding of development, but also aid our understanding of evolution. Finally, ex vivo systems or stem cell-based organoids hold enormous potential, as illustrated by their application to study somite development and eye cup formation, where organoids from mice and human recapitulate the species-specific size and timing differences (Ader and Tanaka, 2014; Diaz-Cuadros et al., 2019 preprint; Matsuda et al., 2019 preprint; Nakajima et al., 2018).

We are grateful to Katherine W. Rogers and Alexandra Schauer for helpful discussions.